Let \(X=\operatorname{\mathrm {Hilb}}_d(\text{\bf P}^n)\)  and \(Y=\operatorname{\mathrm {Hilb}}_d(\mathbf {A}^n,0)\) .
The idea is to use the action of the multiplicative
group \(T\)  (that is, \(\text{\bf C}^*\) ) on \(\operatorname{\mathrm {Hilb}}_d(\text{\bf P}^n)\) ,
coming from the action
of \(T\)  on \(\text{\bf P}^n\)  by
\(t([x_0,\ldots ,x_n])=[x_0,tx_1,\ldots ,tx_n].\) 
(We identify \(\mathbf {A}^n\)  with the open subset \(x_0\ne 0\)  in \(\text{\bf P}^n\) .)
Here \(X\)  has a
\(\operatorname{GL}(n+1)\) -equivariant ample line bundle by construction.
(Namely, Grothendieck constructed the Hilbert scheme
as a closed subscheme of the Grassmannian
of subspaces of the vector space
of homogeneous polynomials of sufficiently high
degree, sending a closed subscheme \(S\subset \text{\bf P}^n\)  to the linear subspace
of polynomials that vanish on \(S\)  [1]}.
The standard ample line bundle \(O(1)\)  on the Grassmannian
is \(GL(n+1)\) -equivariant.)
In particular, \(X\)  has
a \(T\) -equivariant ample line bundle.
The open subset \(U\subset X\)  of 0-dimensional schemes
that converge as \(t\in T\)  approaches 0 to a subscheme supported
at \([1,0,\ldots ,0]\)  is exactly \(\operatorname{\mathrm {Hilb}}_d(\mathbf {A}^n)\) .
The action of \(T\)  on \(U\)  need not extend to a morphism \(\mathbf {A}^1\times U\rightarrow U\) 
(or even \(\mathbf {A}^1\times Y\rightarrow Y\) ). Nonetheless, the desired
homotopy equivalence follows from Theorem REF .
